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Some Useful Spherically Symmetric Profiles Used to Model Galaxies

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Some Useful Spherically Symmetric Profiles Used to Model Galaxies Some useful spherically symmetric profiles used to model galaxies de Vaucouleurs’ R1/4 law for ellipticals and bulges A good fit to the light profile of many ellipticals and bulges: (constant such that Re encloses 1/2 light) I = surface brightness R = projected radius Re = effective radius enclosing 1/2 of light M87 Sérsic profile Generalization of de Vaucouleurs’ R1/4 law n=Sérsic index (=4 for R1/4) More luminous galaxies tend to have larger n n=1 gives exponential profile which is a good approximation to many spirals 1990ApJ...356..359H 1990ApJ...356..359H Hernquist profile Sky projection is close to R1/4 law, but 3D mass distribution is de Vaucouleurs analytically tractable (unlike R1/4): Hernquist M = total mass a = scale radius Convenient for theoretical models of ellipticals and bulges Hernquist 90 Cosmological N-body simulations and dark matter halos The standard Λ Cold Dark Matter cosmology Ωdm Ωb ΩΛ Combined with other astronomical ‣ spectrum of initial density fluctuations measurements, the cosmic microwave ‣ what the Universe is made out of background tells us: ‣ how old it is and how it has expanded in time Initial conditions for cosmological simulations Microwave sky Simulation ICs 1x 4x 64x 16x Visualization credit: Hahn Density field in a 100 Mpc/h box with two initial levels of Example of an N-body simulation of a deeply nested re- refinement generated with MUSIC. gion of 6 initial levels generated with MUSIC and evolved Gaussian random field filtered with transfer function to with Gadget-2 to achieve an effective resolution of 81923 model early Universe physics (photon-baryon interactions) with 11603 particles in the high-res region. Introduction Description of the method MUSIC (MUlti-Scale Initial Conditions) generates cosmological initial conditions for a hierarchical set of nested regions. A detailed description of the method can be found in the code paper Hahn&Abel (2011), http://arxiv.org/abs/1103.6031. We kindly refer the reader to that paper for all technical aspects as well as performance and validation of the code. Cookbook for setting up a zoom simulation with MUSIC The procedure for setting up a zoom simulation follows typically the procedure of 4 steps, given below. Note that resolution levels in MUSIC are specified by their linear power-2 exponent, i.e. a resolution of 1283 cells or particles corresponds to level 7 (log2 128=7). We use the term “lower” for levels synonymously with “coarser” and “higher” with “finer”. Run a unigrid dark matter-only pre-flight simulation In order to set up unigrid initial conditions with MUSIC, select first the desired resolution for this pre-flight simulation. 3 Assume we want to run a 128 simulation, the coarse grid level has to be set to log2 128=7. Since we want to run a unigrid simulation, both levelmin and levelmax in section [setup] should be set to 7. Also the coarse grid seed needs to be chosen now and must not be changed afterwards. To do this, we set seed[7] in section [random] to the desired random seed. This seed determines the large scale structure and we will only add subgrid noise when performing refinement later. Now, set the box size, starting redshift, all the cosmological parameters and the input transfer function in the respective sections. Finally, select the output plugin in section [output] for the code with which you wish to perform this pre-flight simulation. Finally run MUSIC with the configuration file that contains all your settings and start your simulation. Note that you also have to explicitly specify a redshift at which to generate the initial conditions. MUSIC - User's Manual 3 What is smoothed particle hydrodynamics? DIFFERENT METHODS TO DISCRETIZE A FLUID What is smoothed particle hydrodynamics? EulerianDIFFERENT METHODS TO DISCRETIZE ALagrangian FLUID discretize space Eulerian discretize massLagrangian representation on a mesh representation by fluid elements (volume elements)discretize space (particles) discretize mass representation onN a-body mesh simulations representation by fluid elements (volume elements) (particles) • Discretize mass with N particles ‣ in cosmology, usually tree or particle- mesh methods to solve Poisson’s equation principle advantage: principle advantage: high accuracy (shock capturing), low resolutions adjusts Naturally adaptive in cosmology numerical viscosityprinciple advantage:• automatically to theprinciple flow advantage: high accuracy (shock capturing), low resolutions adjusts numerical viscosity automatically to the flow collapse collapse History of N-body simulations Collisionless (cosmological) Moore’s law Collisional (e.g., globulars) Dehnen & Read 11 Gravity amplifies primordial fluctuations, forms structures Simplest: dark matter only ~100 Mpc Visualization credit: Kravtsov Density peaks (dark matter halos) are the sites of galaxy formation Millennium simulation (z=0 fly through) 1010 particles, 500 h-1 Mpc Springel+05 NATURE|Vol 435|2 June 2005 ARTICLES based on this formula may contain large errors15. We return below to black holes with a mass a billion times that of the Sun. At redshift the important question of the abundance of quasars at early times. z < 6, their co-moving space density is estimated to be To track the formation of galaxies and quasars in the simulation, ,(2.2 ^ 0.73) 1029h 3 Mpc23 (ref. 25). Whether such extremely we implement a semi-analytic model to follow gas, star and super- rare objects can£ form at all in a LCDM cosmology is unknown. massive black-hole processes within the merger history trees of dark A volume the size of the Millennium Simulation should contain, matter haloes and their substructures (see Supplementary Infor- on average, just under one quasar at the above space density. Just mation). The trees contain a total of about 800 million nodes, each what sort of object should be associated with these ‘first quasars’ is, corresponding to a dark matter subhalo and its associated galaxies. however, a matter of debate. In the local Universe, it appears that This methodology allows us to test, during postprocessing, many every bright galaxy hosts a supermassive black hole and there is a different phenomenological treatments of gas cooling, star for- remarkably good correlation between the mass of the central black mation, AGN growth, feedback, chemical enrichment and so on. hole and the stellar mass or velocity dispersion of the bulge of the Here, we use an update of models described in refs 16 and 17, which host galaxy26. It would therefore seem natural to assume that, at any are similar in spirit to previous semi-analytic models18–23;the epoch, the brightest quasars are always hosted by the largest galaxies. modelling assumptions and parameters are adjusted by trial and In our simulation, ‘large galaxies’ can be identified in various ways, error to fit the observed properties of low-redshift galaxies, primarily for example, according to their dark matter halo mass, stellar mass or their joint luminosity–colour distribution and their distributions of instantaneous star-formation rate. We have identified the ten ‘largest’ morphology, gas content and central black-hole mass. Our use of a objects defined in these three ways at redshift z 6.2. It turns out high-resolution simulation, particularly our ability to track the that these criteria all select essentially the same¼ objects: the eight evolution of dark matter substructures, removes much of the largest galaxies by halo mass are identical to the eight largest galaxies uncertainty of the more traditional semi-analytic approaches based by stellar mass; only the ranking differs. Somewhat larger differences on Monte Carlo realizations of merger trees. Our technique provides are present when galaxies are selected by star-formation rate, but accurate positions and peculiar velocities for all the model galaxies. It the four first-ranked galaxies are still among the eight identified also enables us to follow the evolutionary history of individual according to the other two criteria. objects and thus to investigate the relationship between populations In Fig. 3, we illustrate the environment of a ‘first quasar’ candidate seen at different epochs. It is the ability to establish such evolutionary in our simulation at z 6.2. The object lies on one of the most connections that makes this kind of modelling so powerful for prominent dark matter¼ filaments and is surrounded by a large interpreting observational data. number of other, much fainter galaxies. It has a stellar mass of 10 21 6.8 10 h M (, the largest in the entire simulation at z 6.2, a £ 12 21 ¼ The fate of the first quasars dark matter virial mass of 3.9 10 h M (, and a star-formation 21 £ Quasars are among the most luminous objects in the Universe and rate of 235M ( yr . In the local Universe, central black-hole masses can be detected at huge cosmological distances. Their luminosity is are typically ,1/1,000 of the bulge stellar mass27, but in the model we thought to be powered by accretion onto a central, supermassive test here these massive early galaxies have black-hole masses in the 8 9 black hole. Bright quasars have now been discovered as far back range 10 –10 M (, significantly larger than low-redshift galaxies of as redshift z 6.43 (ref. 24), and are believed to harbour central similar stellar mass. To attain the observed luminosities, they must Dark matter¼ halo mass functionconvert infalling mass to radiated energy with a somewhat higher efficiency than the ,0.1c 2 expected for accretion onto a non- spinning black hole (where c is the speed of light in vacuum). #Within halos our simulationper volume we can readilyper addressmass fundamental ques- tions such as: Where are the descendants of the early quasars today? Whatinterval were their progenitors? By tracking the merging history trees of the host haloes, we find that all our quasar candidates end up today as central galaxies in rich clusters.
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